#7613491

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,23)(13,27)(14,18)(15,20)(16,26)(17,28)(21,32)(22,34)(24,29)(25,39)(30,36)(31,37)(33,35)(38,40);(2,5)(6,8)(11,20)(13,28)(15,19)(16,31)(17,27)(21,35)(22,36)(24,40)(26,37)(29,38)(30,34)(32,33);(1,3,4,7)(2,6,5,8)(9,11,13,20,14,22,28,36)(10,15,17,19,18,30,27,34)(12,24,26,40,25,21,37,35)(16,32,23,38,31,29,39,33);(9,12,14,25)(10,16)(11,21,22,24)(13,26,28,37)(15,29)(17,23)(18,31)(19,33)(20,35,36,40)(27,39)(30,32)(34,38);(2,5)(6,8)(12,25)(16,31)(21,24)(23,39)(26,37)(29,32)(33,38)(35,40)

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520687);